Define Scalar product of two vector. Give any four properties of scalar product.

Explanation

(i) The scalar product (or dot product) of two vectors is defined as the product of the magnitudes of both the vectors and the cosine of the angle between them.
(ii) Thus if there are two vectors `vec(A) " and " vec(B)` having an angle `theta` between them, then their scalar product is defined as `vec(A)*vec(B)=ABcostheta`. Here, A and B are magnitudes of `vec(A) " and " vec(B)`.
Properties of Scalar Products:
(i) The product quantity `vec(A).vec(B)` is always a scalar. It is positive if the angle between the vectors is acute (i.e. `theta lt 90^(@)`) and negative if the angle between them is obtuse
(i.e. `90^(@) lt theta lt 180^(@)`).
(ii) The scalar product is commutative,
i.e. `vec(A)*vec(B)=vec(B)*vec(A)`
(iii) The vectors obey distributive law i.e.
`vec(A)*(vec(B)+vec(C))=vec(A).vec(B)+vec(A).vec(C)`
(iv) The angle between the vectors
`theta=cos^(-1)[(vec(A).vec(B))/(AB)]`